First principles study of point defects in titanium oxycarbide
H. M. Pintoa , J. Coutinhob, M. M. D. Ramosa , F. Vaza , L. Marquesa,∗
a University
of Minho, Department of Physics, Campus de Gualtar, 4710-057 Braga, Portugal
of Aveiro, Department of Physics, Campus Santiago, 3810-193 Aveiro, Portugal
b University
Abstract
We have performed first principles density functional theory calculations to study the formation energy of point defects
in TiC, TiO and TiCO compounds. The formation energy of isolated vacancies were obtained for different equilibrium
conditions. For binary compounds, we have also calculated the formation energy of antisite defects. It was found that
the defect formation energies strongly depend on the chemical environment. Our results show that C vacancies are
easily formed in TiC and TiCO. For the TiO compound, Ti vacancies are highly probable to occur and O vacancies
are also easily formed under titanium rich atmosphere.
Key words: Defect formation, Titanium oxide, Titanium carbide
1. Introduction
Titanium oxycarbide is a multifunctional material crystalizing in the rocksalt structure. Changing the oxygen/carbon ratio it is possible to tune its mechanical and electrical properties from the metallic-like, such has high
electrical conductivity and ductility to insulating behavior and extreme hardness of typical covalent bonded materials.
[1, 2, 3] This variety of properties has attracted the attention to this material due to its potential for a wide range of
high technological applications. These materials presents a high concentration of defects, specially vacancies, which
play an important role in the material properties.[5, 4] Thus, it is important to investigate the formation of isolated
vacancies and antisite point defects in titanium oxycarbide, which can be obtained at different chemical environment.
In this work we present a theoretical study of point defects in titanium oxycarbides using first principles calculations. The formation energy of vacancies and antisite defects for the binary TiC and TiO compounds and the formation
energy of vacancies in the ternary compound TiC 0.5 O0.5 are calculated by ab initio methods.
2. Method
The calculations were performed within the density functional theory (DFT), using the (VASP) code [7, 8, 9],
with local density approximation (LDA) for the exchange-correlation energy and projector augmented wave (PAW)
∗
Email addresses: [email protected] (L. Marques)
Preprint submitted to Plasma Processes & Polymers
February 3, 2009
pseudopotentials [10], in supercells containing 64 atoms. Cut-off energy (E cut ) of 500 eV was used for the plane-wave
basis set expansion. The Brillouin zone (BZ) was sampled using the Monkhorst-Pack scheme [11] with a mesh of
7×7×7 points in the reciprocal unit cell.
The formation energy of a defect in a binary compound like TiC is given by [12]:
ΩD = ED − nTi µTi − nC µC
(1)
where ED is the total energy of a supercell containing n Ti Ti atoms, nC C atoms and one defect, µ Ti and µC are the
chemical potentials of Ti and C, respectively. This equation may be written in the form
1
ΩD = ED − (nC − nTi )Δµ
2
(2)
1
1
ED = ED − (nC + nTi )µBTiC − (nC − nTi )(µCB − µBTi )
2
2
(3)
where
The superscript B in the chemical potential indicates the corresponding bulk value.The difference Δµ is defined as
Δµ = (µC − µTi ) − (µCB − µBTi )
(4)
The chemical potential of Ti and C should not exceed their bulk chemical potential, otherwise the respective precipitate
would be formed. Consequently, µ Ti µBTi and µC µCB . At thermal equilibrium, the sum of the Ti and C chemical
potentials is equal to the chemical potential of bulk TiC per pair of atoms; µ Ti + µC = µBTiC . Then, the difference in the
chemical potentials of both elements is restricted by the following condition
−ΔH Δµ ΔH
(5)
where ΔH is the heat of formation of TiC. This determines the growth conditions during the defect formation: Δµ = ΔH
corresponds to Ti-rich limit and Δµ = −ΔH corresponds to C-rich limit. The bulk chemical potentials of Ti, C, O,
TiC and TiO were calculated using VASP, assuming a hcp structure in case of titanium, the diamond structure for
carbon and a NACl structure for TiC and TiO compounds. For the calculation of the chemical potential of oxygen we
considered a large supercell with an oxygen molecule in the center.
In order to illustrate the application of the theoretical formalism described to ternary compounds, we will just consider in this work, as a reference, an ordered TiCO alloy with the composition TiC 0.5 O0.5 and the spatial arrangement
illustrated in figure 1 for an eight atom supercell. In the case of a ternary compound, like TiCO, the defect formation
energy is given by [13]
ΩD = ED − nTi µTi − nC µC − nO µO
(6)
The Ti, C and O chemical potencials (µ Ti , µC e µO ) should not exceed their corresponding bulk values and the sum of
those chemical potentials must satisfy the following conditions
µTi + µC µBTiC
2
(7)
Figure 1: NaCl structure of TiC0.5 C0.5 . One sublattice is occupied by metal atoms (grey) and the other by carbon (black) and oxygen atoms (white),
with the spatial distributions shown.
and
µTi + µO µBTiO
(8)
At thermal equilibrium, the sum of the chemical potentials of the isolated elements is equal to to the chemical potential
of the ternary compound,
2µTi + µC + µO µBTiC0.5 O0.5
(9)
This conditions limit the ranges over which the chemical potentials can vary. Defining two new parameters Δµ and δµ
Δµ = (µO + µC − 2µTi ) − (µBO + µCB − 2µBTi )
(10)
δµ = (µC − µO ) − (µCB − µBO )
(11)
and
the formation energy in Eq. 6 can be rewritten as
1
1
ΩD = ED − (nC + nO − nTi )Δµ − (nC − nO )δµ
4
2
(12)
where
1
nTi B
1
nTi B
ED = ED − (nC −
)(µC − µBTi ) − (nO −
)(µO − µBTi )
2
2
2
2
1
1
− (nC + nO + nTi )µBTiC0.5 O0.5 − (nC − nO )(µCB − µBO )
4
4
(13)
which only depends of the bulk chemical potencial of Ti, C and O. On the other hand, the formation energy Ω D only
depends on Δµ and δµ. In the case of the ternary compound, we have
−ΔH Δµ ΔH
3
(14)
Figure 2: The Ti- (A→B), C- (B→C) and O-rich (C→A) limits are represented as function of Δµ and δµ, which change along the range bounded
by the triangle.
where
ΔH = (µBO + µCB + 2µBTi ) − µBTiC0.5 O0.5
(15)
The parameter δµ is restricted by the constraint
1
1
− (ΔH − Δµ) δµ (ΔH − Δµ)
2
2
(16)
In the Ti-rich condition (Δµ = −ΔH), δµ varies from −ΔH (O-rich limit) to ΔH (C-rich limit). In O- and C- rich
conditions (Δµ = ΔH) thus δµ is equal to zero. Figure 2 shows the different chemical environments defined by Δµ and
δµ. Note that Δµ has a minimum value at Ti-rich conditions and a maximum at Ti-poor conditions.
The formalism presented in this section, for binary and ternary alloys, was used to predict the relative occurrence
probability of several types of structural single point defects in TiC, TiO and TiC 0.5 O0.5 compounds for different
chemical growth conditions.
3. Results and Discussion
3.1. Vacancies and antisites in TiC and TiO
Table 1 presents the vacancy (V Ti and VC ) and antisite (TiC and CTi ) formation energies for TiC compound. The
formation energies of all defects are also plotted in figure 3 for the different equilibrium conditions.
The VC is found to be the most probable defect in TiC, which is in agreement with experiments [4]. In Ti-rich
environment the VC has a negative formation energy meaning that under this condition carbon vacancies are created
spontaneously. However, this may happen only when enough thermal energy is available to surmount an eventual
potential barrier. The other defects are very unlikely to occur because their formation requires high energies.
The calculated formation energies of vacancies (V Ti , VO ) and antisite (TiO , OTi ) defects for TiO are presented in
the table 2 and plotted in Figure 4 for the different chemical environments.
4
VTi
VC
TiC
CTi
Ti-rich
9.34
-0.95
5.88
14.90
C-rich
7.49
0.89
9.58
11.20
Table 1: Formation energies (ΩD ) in eV, of vacancies (VTi , VC ) and antisites (TiC , CTi ) in TiC.
Figure 3: Defect formation energies (ΩD ) as function of chemical potentials from Ti-rich limit to the C-rich limit, for TiC.
VTi
VO
TiO
OTi
Ti-rich
-0.74
-1.37
6.74
7.75
O-rich
-6.82
4.71
18.91
-4.41
Table 2: Formation energies (ΩD ) in eV, of vacancies (VTi , VO ) and antisites (TiO , OTi ) in TiO.
Under Ti-rich conditions the V Ti and VO formation energies are both negative and are highly probable to occur
with formation energy of -0.74 and -1.37 respectively. This results are supported by the experimental observation
of vacancies in both lattices of TiO compounds [5, 14]. The antisites are unlikely to occur in Ti-rich conditions
due to their high formation energies. Oxidation conditions are favorable to the formation of O antisites (O Ti ) and
Ti vacancy defects (V Ti ) in the metallic lattice, due to their negative formation energies. This suggests an imminent
phase transition towards a substoichiometric TiO 2 . The relatively high formation energy of V O and TiO defects means
that under O-rich conditions they are hard to be formed in TiO.
3.2. Vacancies in TiC 0.5 O0.5
The formation energy of vacancies was studied for the ternary compound TiC 0.5 O0.5 as function of the chemical
potential difference (see figure 2). Table 3 presents values obtained for the formation energies of Ti, C and O vacancies
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Figure 4: Defect formation energies as function of atomic chemical potentials between two extremes, the Ti-rich and O-rich conditions, for TiO.
(VTi , VC and VO ) in TiC0.5 O0.5 , for the different chemical environments.
Ti-, O-rich
Ti-, C-rich
C-, O-rich
VTi
2.40
2.40
-1.80
VC
-7.59
0.82
-7.59
VO
6.10
-2.31
6.10
C-poor
O-poor
Ti-poor
Table 3: Formation energies (ΩD ) of vacancies (VTi , VC and VO ) for TiC0.5 O0.5
In Ti-rich and C-poor conditions (A) the carbon vacancy (V C ) is highly probable to occur. In this conditions the
other defects, (V Ti and VO ) hardly occur due to their high formation energies. In C-rich and O-poor conditions (B),
VO has the lowest formation energy and V Ti has the highest one. These are the unique conditions favorable to the
occurrence of V O defect in TiC0.5 O0.5 . For O-rich and Ti-poor conditions (C), C and Ti vacancies are likely to occur.
As expected, in O-rich atmospheres the occurrence of O vacancies are unlikely to occur.
4. Conclusions
This work provides a general overview of the formation of several types of structural point defects in titanium
oxycarbides for different chemical growth conditions. We used first principles methods to calculate the formation
energies of vacancies and antisite defects in TiC, TiO compounds and the formation energy of vacancies in a ordered
TiCO alloy with composition TiC 0.5 O0.5 . Our calculations suggest that C vacancies are likely to be formed during
the growth of TiC and TiC 0.5 O0.5 , whereas Ti vacancies are highly probable to occur during the growth of TiO and O
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Figure 5: The defect formation energies are plotted in the Ti-, C-, and O-rich limits for TiC0.5 O0.5 .
antisite defects are also likely to be formed on O-rich atmospheres leading to an imminent phase transition towards
a substoichiometric TiO 2 . The obtained results are supported by the existent experimental studies of point defects in
TiC and TiO compounds.
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